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Math 129 Final Exam Fall 202011/18/2020 (online, submitted on Gradescope)Directions. (Please read!)• You must show all your work and justify your methods to obtain full credit. Name anytheorems that you use. Clearly indicate your final answers.• Simplify your answers to a reasonable degree.• You may use your personal notes, your book and a calculator during the test. You can use yourcell phone only at the end of the test to take pictures of your paper and submit. Collaborationwith other students/external people is strictly forbidden.• Your camera should be on and your microphone muted at all times during the test. If youexperience a technical issue, please let your instructor know immediately. For any questionto your instructor/TA, please use the chat.• The test lasts two hours. You will have additional 20 minutes to submit on Gradescope.Please use the extra time exclusively for submission. Late submissions will be accepted onlyunder exceptional circumstances.• Please indicate the question number/part you are working on on your paper. Once you aredone, you should submit one pdf on Gradescope. Please match the pages with the questionnumbers on the outline.• Remember, USC considers cheating to be a serious offense; the minimum penalty is failure forthe course. In the current circumstances, cheating includes communication with your peersvia email, social media, etc.Problem 1. Find the following limits if they exist (including ±∞). In case they do not exist,explain why.a) limx→0e2x− 1tan−1(x).b) limx→0√1 + 2x − x − 1x2.c) limx→∞xπ2− arctan x.(20 pt.)Problem 2. Evaluate the following integrals:a)Z√x sin√x dx.b)Zsin3(x) cos(3x) dx. Hint: cos(3x) = 4 cos3(x) − 3 cos(x).c)Zx2(9 − x2)3/2dx.(20 pt.)Problem 3. a) Determine whether the improper integralZ∞0xe−2xdxconverges or diverges by carefully evaluating it.b) Determine whetherZ∞81x15/7− x1/π− 1√x9+ x6+ 27dx converges or diverges, using any valid argument.(20 pt.)Problem 4. The region R is bounded by x = yeyand y =1ex. Find the volume obtained byrotating R about the y-axis.(20 pt.)Problem 5. A planter in the shape of a trapezoidal prism (see picture below) is filled with water.The bottom of the planter is a 1m by 3m rectangle, and the front and back faces are isoscelestrapezoids. Both trapezoids have bases of 1m and 2m, and a height of 1m. Recall the density ofwater is ρ = 1000 kg/m3and g = 9.8 m/s2.(a) SET UP, but DO NOT EVALUATE, an integral representing the hydrostatic force on thefront face of the planter.(b) SET UP, but DO NOT EVALUATE, an integral representing the work required to pump allthe water out of the top of the planter.(20 pt.)Problem 6. Consider the power series∞Xn=1n!rnnn(x − 3)n,where r ≥ 0.a) Find its radius of convergence as a function of r (Hint. limn→∞(1 + 1/n)n= e);b) Find a value of r ≥ 0 such that the series convergent at x = 4 and divergent at x = 0.(20 pt.)Problem 7. Determine whether the given numerical series are convergent or divergent. Justifyyour answer in each case.a)∞Xn=1ln5n2n√nb)∞Xn=11n tan(1/n)(20 pt.)Problem 8. A wire is bent into a circle of radius R and a current is run through it. It exerts aforce on a positively charged particle located distance x above the center of the circle given byf(x) =−qx(x2+ R2)3/2where q > 0 is a constant.a) Find the Taylor series of f(x) centered at a = 0.b) Find the Taylor polynomial of degree 3 for the above series.c) Use the above Taylor polynomial to estimate f(R/2).d) Estimate the error in the above approximation.(20

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